Existence of Bertrand equilibrium revisited

In this paper we reconsider existence of Bertrand equilibrium in a symmetric-cost, homogenous-product oligopoly. We prove the following main results. (a) If the cost function is strictly superadditive on [0, 8) then there exists a pure strategy Bertrand equilibrium. Such Bertrand equilibria are necessarily non-unique. (b) If the cost function is strictly subadditive on [0, 8) then there exists no Bertrand equilibrium, either in pure strategies or in mixed strategies.